270 Proceedings of Royal Society of Edinburgh. [sess. 
(Rxx - s n) x n + h. xy y n + A xz Z n + * ' * =0 
A xy X n (A yy Cll)y n 4 " A yiZ n ■+* ' =0 
A xz X n + KyzVn + (A zz-S n )z n + • • ■ =0 
xf + y 2 + z 2 + • • • = 1 
•and that, further, when r and s are unequal 
x r x s + y r y s + z r z s + • • = 0 . 
Recalling this, Cauchy says that if a new set of n variables be 
taken 
£ 5 V » £ 5 • • • • 
related to the old by the equations 
X = xf "4* X^X] -J- ~h ' ' ' 
y = y i£ + vw + y£ + • • • 
z M zf + z 2 r] + z 3 £ + • • • 
it is at once verifiable that 
x 2 + y 2 + z 2 + • • • = £ 2 + ^ + i» + . . . 
In the second place, if we take any one, say the first of the set of 
•equations connecting , x x , y 1 , z lt ... , the corresponding 
equation of the set connecting s 2 , x 2 , y 2 , z 2 , ... , and so forth, 
writing them in the form 
A^^q + A xy y^ + A X 7 z^ + • • • = s^x^ , 
Rxx X 2 + A ^2 + A xz Z 2 + • • • = S 2 £ 2 , 
multiplication by £ , g , £ , ... respectively, followed by addition, 
gives 
A xx x + A xy y + A x yz + • • • = s x x^ + s 9 x 2 tj + s 3 x 3 £ + • • • \ 
A xy x + A yy y + A yz z + • • • = s lV f + s 2 y 2 rj + s 3 y 3 C + f 
A xz x + A,; Jz y + A zz z + • • • - sfat + s^ 2 rj + s 3 z 3 £ + • f 
In the third place if the equations giving x , y , z , ... in terms 
-of £ , y , t , ... be taken, multiplication by ir r , , . . . respec- 
tively, followed by addition, gives 
